To solve triangle $ABC$ with angles and side lengths given as follows:

• $B = 67^\circ$
• $C = 35^\circ$
• $c = 62$

Step 1: Find Angle $A$

Since the sum of angles in a triangle is $180^\circ$, we can find $A$ using: $A = 180^\circ - B - C$ $A = 180^\circ - 67^\circ - 35^\circ = 78^\circ$

So, $A = 78^\circ$.

Step 2: Use the Law of Sines to Find Sides $a$ and $b$

The Law of Sines states: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Solving for $a$

$\frac{a}{\sin 78^\circ} = \frac{62}{\sin 35^\circ}$ Rearrange to solve for $a$: $a = 62 \times \frac{\sin 78^\circ}{\sin 35^\circ}$ Calculating $a$: $a \approx 62 \times \frac{0.9781}{0.5736} \approx 105.7$

Solving for $b$

$\frac{b}{\sin 67^\circ} = \frac{62}{\sin 35^\circ}$ Rearrange to solve for $b$: $b = 62 \times \frac{\sin 67^\circ}{\sin 35^\circ}$ Calculating $b$: $b \approx 62 \times \frac{0.9205}{0.5736} \approx 99.5$

Summary of Results

• $A = 78^\circ$
• $a \approx 105.7$
• $b \approx 99.5$

Would you like further details or have any questions?

Here are 5 related questions:

1. How is the Law of Sines applied in different types of triangles?
2. What are alternative methods for solving non-right triangles?
3. How does the Law of Sines compare to the Law of Cosines?
4. What happens when there is an ambiguous case in the Law of Sines?
5. How would the calculations change if the given angle measurements were altered?

Tip: Always check if a triangle has two possible solutions when solving with the Law of Sines, especially when given angle-side-angle configurations.